theorem Th10:
  for a being Element of V1, F being FinSequence of INT.Ring,
  G being FinSequence of V1 st len F = len G &
  for k st k in dom F holds G.k = (F/.k) * a
  holds Sum(G) = Sum(F) * a
  proof
    let a be Element of V1;
    let F be FinSequence of INT.Ring;
    let G be FinSequence of V1;
    assume that
    A1: len F = len G and
    A2: for k st k in dom F holds G.k = (F/.k) * a;
    now
      let k;
      let v be Element of INT.Ring;
      assume that
      A3: k in dom G and
      A4: v = F.k;
      A5: k in dom F by A1,A3,FINSEQ_3:29;
      then v = F/.k by A4,PARTFUN1:def 6;
      hence G.k = v * a by A2,A5;
    end;
    hence thesis by A1,Th9;
  end;
